Principles of Fourier Analysis, 2nd Edition

Principles of Fourier Analysis, 2nd Edition

by Kenneth B. Howell
Released December 2016
Publisher(s): CRC Press
ISBN: 9781498734134

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Book description

Fourier analysis is one of the most useful and widely employed sets of tools for the engineer, the scientist, and the applied mathematician. As such, students and practitioners in these disciplines need a practical and mathematically solid introduction to its principles. They need straightforward verifications of its results and formulas, and they need clear indications of the limitations of those results and formulas.

Principles of Fourier Analysis furnishes all this and more. It provides a comprehensive overview of the mathematical theory of Fourier analysis, including the development of Fourier series, "classical" Fourier transforms, generalized Fourier transforms and analysis, and the discrete theory. Much of the author's development is strikingly different from typical presentations. His approach to defining the classical Fourier transform results in a much cleaner, more coherent theory that leads naturally to a starting point for the generalized theory. He also introduces a new generalized theory based on the use of Gaussian test functions that yields an even more general -yet simpler -theory than usually presented.

Principles of Fourier Analysis stimulates the appreciation and understanding of the fundamental concepts and serves both beginning students who have seen little or no Fourier analysis as well as the more advanced students who need a deeper understanding. Insightful, non-rigorous derivations motivate much of the material, and thought-provoking examples illustrate what can go wrong when formulas are misused. With clear, engaging exposition, readers develop the ability to intelligently handle the more sophisticated mathematics that Fourier analysis ultimately requires.

Table of contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright Page
  5. Table of Contents
  6. Preface
  7. Sample Courses
  8. I Preliminaries
    1. 1 The Starting Point
      1. 1.1 Fourier’s Bold Conjecture
      2. 1.2 Mathematical Preliminaries and the Following Chapters
      3. Additional Exercises
    2. 2 Basic Terminology, Notation and Conventions
      1. 2.1 Numbers
      2. 2.2 Functions, Formulas and Variables
      3. 2.3 Operators and Transforms
    3. 3 Basic Analysis I: Continuity and Smoothness
      1. 3.1 (Dis)Continuity
      2. 3.2 Differentiation
      3. 3.3 Basic Manipulations and Smoothness
      4. 3.4 Addenda
      5. Additional Exercises
    4. 4 Basic Analysis II: Integration and Infinite Series
      1. 4.1 Integration
      2. 4.2 Infinite Series (Summations)
      3. Additional Exercises
    5. 5 Symmetry and Periodicity
      1. 5.1 Even and Odd Functions
      2. 5.2 Periodic Functions
      3. 5.3 Sines and Cosines
      4. Additional Exercises
    6. 6 Elementary Complex Analysis
      1. 6.1 Complex Numbers
      2. 6.2 Complex-Valued Functions
      3. 6.3 The Complex Exponential
      4. 6.4 Functions of a Complex Variable
      5. Additional Exercises
    7. 7 Functions of Several Variables
      1. 7.1 Basic Extensions
      2. 7.2 Single Integrals of Functions with Two Variables
      3. 7.3 Double Integrals
      4. 7.4 Addenda: Proving Theorems 7.7 and 7.9
      5. Additional Exercises
  9. II Fourier Series
    1. 8 Heuristic Derivation of the Fourier Series Formulas
      1. 8.1 The Frequencies
      2. 8.2 The Coefficients
      3. 8.3 Summary
      4. Additional Exercises
    2. 9 The Trigonometric Fourier Series
      1. 9.1 Defining the Trigonometric Fourier Series
      2. 9.2 Computing the Fourier Coefficients
      3. 9.3 Partial Sums and Graphing
      4. Additional Exercises
    3. 10 Fourier Series over Finite Intervals (Sine and Cosine Series)
      1. 10.1 The Basic Fourier Series
      2. 10.2 The Fourier Sine Series
      3. 10.3 The Fourier Cosine Series
      4. 10.4 Using These Series
      5. Additional Exercises
    4. 11 Inner Products, Norms and Orthogonality
      1. 11.1 Inner Products
      2. 11.2 The Norm of a Function
      3. 11.3 Orthogonal Sets of Functions
      4. 11.4 Orthogonal Function Expansions
      5. 11.5 The Schwarz Inequality for Inner Products
      6. 11.6 Bessel’s Inequality
      7. Additional Exercises
    5. 12 The Complex Exponential Fourier Series
      1. 12.1 Derivation
      2. 12.2 Notation and Terminology
      3. 12.3 Computing the Coefficients
      4. 12.4 Partial Sums
      5. Additional Exercises
    6. 13 Convergence and Fourier’s Conjecture
      1. 13.1 Pointwise Convergence
      2. 13.2 Uniform and Nonuniform Approximations
      3. 13.3 Convergence in Norm
      4. 13.4 The Sine and Cosine Series
      5. Additional Exercises
    7. 14 Convergence and Fourier’s Conjecture: The Proofs
      1. 14.1 Basic Theorem on Pointwise Convergence
      2. 14.2 Convergence for a Particular Saw Function
      3. 14.3 Convergence for Arbitrary Saw Functions
      4. 14.4 A Divergent Fourier Series
    8. 15 Derivatives and Integrals of Fourier Series
      1. 15.1 Differentiation of Fourier Series
      2. 15.2 Differentiability and Convergence
      3. 15.3 Integrating Periodic Functions and Fourier Series
      4. 15.4 Sine and Cosine Series
      5. Additional Exercises
    9. 16 Applications
      1. 16.1 The Heat Flow Problem
      2. 16.2 The Vibrating String Problem
      3. 16.3 Functions Defined by Infinite Series
      4. 16.4 Verifying the Heat Flow Problem Solution
      5. Additional Exercises
  10. III Classical Fourier Transforms
    1. 17 Heuristic Derivation of the Classical Fourier Transform
      1. 17.1 Riemann Sums over the Entire Real Line
      2. 17.2 The Derivation
      3. 17.3 Summary
    2. 18 Integrals on Infinite Intervals
      1. 18.1 Absolutely Integrable Functions
      2. 18.2 The Set of Absolutely Integrable Functions
      3. 18.3 Many Useful Facts
      4. 18.4 Functions with Two Variables
      5. Additional Exercises
    3. 19 The Fourier Integral Transforms
      1. 19.1 Definitions, Notation and Terminology
      2. 19.2 Near-Equivalence
      3. 19.3 Linearity
      4. 19.4 Invertibility
      5. 19.5 Other Integral Formulas (A Warning)
      6. 19.6 Some Properties of the Transformed Functions
      7. Additional Exercises
    4. 20 Classical Fourier Transforms and Classically Transformable Functions
      1. 20.1 The First Extension
      2. 20.2 The Set of Classically Transformable Functions
      3. 20.3 The Complete Classical Fourier Transforms
      4. 20.4 What Is and Is Not Classically Transformable?
      5. 20.5 Finite Duration and Finite Bandwidth Functions
      6. 20.6 More on Terminology, Notation and Conventions
      7. Additional Exercises
    5. 21 Some Elementary Identities: Translation, Scaling and Conjugation
      1. 21.1 Translation
      2. 21.2 Scaling
      3. 21.3 Practical Transform Computing
      4. 21.4 Complex Conjugation and Related Symmetries
      5. Additional Exercises
    6. 22 Differentiation and Fourier Transforms
      1. 22.1 The Differentiation Identities
      2. 22.2 Rigorous Derivation of the Differential Identities
      3. 22.3 Higher Order Differential Identities
      4. 22.4 Anti-Differentiation and Integral Identities
      5. Additional Exercises
    7. 23 Gaussians and Gaussian-Like Functions
      1. 23.1 Basic Gaussians
      2. 23.2 General Gaussians
      3. 23.3 Gaussian-Like Functions
      4. Additional Exercises
    8. 24 Convolution and Transforms of Products
      1. 24.1 Derivation of the Convolution Formula
      2. 24.2 Basic Formulas and Properties of Convolution
      3. 24.3 Algebraic Properties
      4. 24.4 Computing Convolutions
      5. 24.5 Existence, Smoothness and Derivatives of Convolutions
      6. 24.6 Convolution and Fourier Analysis
      7. Additional Exercises
    9. 25 Correlation, Square-Integrable Functions and the Fundamental Identity
      1. 25.1 Correlation
      2. 25.2 Square-Integrable/Finite Energy Functions
      3. 25.3 The Fundamental Identity
      4. Additional Exercises
    10. 26 Generalizing the Classical Theory: A Naive Approach
      1. 26.1 Delta Functions
      2. 26.2 Transforms of Periodic Functions
      3. 26.3 Arrays of Delta Functions
      4. 26.4 The Generalized Derivative
      5. Additional Exercises
    11. 27 Fourier Analysis in the Analysis of Systems
      1. 27.1 Linear, Shift-Invariant Systems
      2. 27.2 Computing Outputs for LSI Systems
      3. Additional Exercises
    12. 28 Multi-Dimensional Fourier Transforms
      1. 28.1 Basic Definitions
      2. 28.2 Computing Multi-Dimensional Transforms
      3. Additional Exercises
    13. 29 Identity Sequences
      1. 29.1 An Elementary Identity Sequence
      2. 29.2 General Identity Sequences
      3. 29.3 Gaussian Identity Sequences
      4. 29.4 Verifying Identity Sequences
      5. 29.5 An Application (with Exercises)
      6. 29.6 Laplace Transforms as Fourier Transforms
      7. Additional Exercises
    14. 30 Gaussians as Test Functions and Proofs of Important Theorems
      1. 30.1 Testing for Equality with Gaussians
      2. 30.2 The Fundamental Theorem on Invertibility
      3. 30.3 The Fourier Differential Identities
      4. 30.4 The Fundamental and Convolution Identities of Fourier Analysis
  11. IV Generalized Functions and Fourier Transforms
    1. 31 A Starting Point for the Generalized Theory
      1. 31.1 Starting Points
      2. Additional Exercises
    2. 32 Gaussian Test Functions
      1. 32.1 The Space of Gaussian Test Functions
      2. 32.2 On Using theSpace ofGaussianTestFunctions
      3. 32.3 Other Test Function Spaces and a Confession
      4. 32.4 More on Gaussian Test Functions
      5. 32.5 Norms and Operational Continuity
      6. Additional Exercises
    3. 33 Generalized Functions
      1. 33.1 Functionals
      2. 33.2 Generalized Functions
      3. 33.3 Basic Algebra of Generalized Functions
      4. 33.4 Generalized Functions Based on Other Test Function Spaces
      5. 33.5 Some Consequences of Functional Continuity
      6. 33.6 The Details of Functional Continuity
      7. Additional Exercises
    4. 34 Sequences and Series of Generalized Functions
      1. 34.1 Sequences and Limits
      2. 34.2 Infinite Series (Summations)
      3. 34.3 A Little More on Delta Functions
      4. 34.4 Arrays of Delta Functions
      5. Additional Exercises
    5. 35 Basic Transforms of Generalized Fourier Analysis
      1. 35.1 Fourier Transforms
      2. 35.2 Generalized Scaling of the Variable
      3. 35.3 Generalized Translation/Shifting
      4. 35.4 The Generalized Derivative
      5. 35.5 Transforms of Limits and Series
      6. 35.6 Adjoint-Defined Transforms in General
      7. 35.7 Generalized Complex Conjugation
      8. Additional Exercises
    6. 36 Generalized Products, Convolutions and Definite Integrals
      1. 36.1 Multiplication and Convolution
      2. 36.2 Definite Integrals of Generalized Functions
      3. 36.3 Appendix: On Defining Generalized Products and Convolutions
      4. Additional Exercises
    7. 37 Periodic Functions and Regular Arrays
      1. 37.1 Periodic Generalized Functions
      2. 37.2 Fourier Series for Periodic Generalized Functions
      3. 37.3 On Proving Theorem 37.5
      4. Additional Exercises
    8. 38 Pole Functions and General Solutions to Simple Equations
      1. 38.1 Basics on Solving Simple Algebraic Equations
      2. 38.2 Homogeneous Equations with Polynomial Factors
      3. 38.3 Nonhomogeneous Equations with Polynomial Factors
      4. 38.4 The Pole Functions
      5. 38.5 Pole Functions in Transforms, Products and Solutions
      6. Additional Exercises
  12. V The Discrete Theory
    1. 39 Periodic, Regular Arrays
      1. 39.1 The Index Period and Other Basic Notions
      2. 39.2 Fourier Series and Transforms of Periodic, Regular Arrays
      3. Additional Exercises
    2. 40 Sampling, Discrete Fourier Transforms and FFTs
      1. 40.1 Some General Conventions and Terminology
      2. 40.2 Sampling and the Discrete Approximation
      3. 40.3 The Discrete Approximation and Its Transforms
      4. 40.4 The Discrete Fourier Transforms
      5. 40.5 Discrete Transform Identities
      6. 40.6 Fast Fourier Transforms
      7. Additional Exercises
  13. Tables, References and Answers
    1. Table A.1: Fourier Transforms of Some Common Functions
    2. Table A.2: Identities for the Fourier Transforms
    3. References
    4. Answers to Selected Exercises
  14. Index

Product information

  • Title: Principles of Fourier Analysis, 2nd Edition
  • Author(s): Kenneth B. Howell
  • Release date: December 2016
  • Publisher(s): CRC Press
  • ISBN: 9781498734134